Why is it that gravitational fields don't decline naturally, according to the law of conservation of energy? If it is the force of gravity which is keeping the Moon in orbit via force-carrying particles, then surely this source of energy should slowly evaporate?

I think it has something to do with current theory which holds that gravitons are massless like the photon. Because both travel at the speed of light, their internal 'clocks' have stopped. They don't experience time as such.

This phrase is being slung about like a cudgel in discussions on this forum lately - and a poorly used cudgel at that.

This law, simply put "states that the total amount of energy in any isolated system remains constant but cannot be recreated, although it may change forms"

Well, so far it has yet to be applied to an isolated system - ANY isolated system. Even an electron is not an isolated system, including the single electrons running around at Cern.

If we are going to use this law as a part of an argument, understand it and apply it correctly. Otherwise, get out a book on physics and try reading.

The earth-moon system is only a part of a greater gravitational system that includes the sun, without which we would all fly off into darkest space, and all the other planets, the Ort cloud, (even minor as the latter are) etc., etc.

In order for an electron to be losing energy you have to look in a region of space around it and show that over some length of time, energy has flowed out of that region of space. An electron sitting still has an electric field around it, but this electric field is "static" (in other words, it isn't flowing in or out of that region). The same logic holds true for the earth just sitting around in space. There is a gravitational field around it, but no flow of energy due to that field, so the earth isn't giving off gravitational energy.

In order for the electron (or the earth) to give off energy, something has to put some energy in in order to make it start moving (i.e. accelerate it). In this case, the energy that gets put in will be partially converted into the energy that is given off--the object itself is not decaying. We see that energy given off as electromagnetic or gravitational waves.

Now if you want to look at photons/gravitons (which is the hard way of doing things here), you should see that for a static field, the number of photons/gravitons moving away from your object will be, on average, the same number as move back towards it, so that on average energy isn't being lost from your object.

If the mythical graviton exists its exists within all known particles held within and not emitted !as to the graviton being massless well again not true is just very very very small beyond our measurement means !

The Earths orbit is changing slightly and measurably as is the moons orbit !